#### Transcript Document

```An equation is a mathematical sentence that contains an = sign.
555 = A-75 A-75 = 555 are examples of an equation.
In the examples above “A” is a variable which represents an unknown
number.
One way to solve for A is to complete the Inverse (opposite) Operation
(the opposite of subtraction is addition)
555 + 75 = A or A = 555 + 75, so A = 630
1. T - 5 = 11
11 + 5 = T
16 = 16
2. 16 - T = 11
Find the example
that does not
use the inverse
Operation.
T = 16 - 11
T=5
3. B + 6 = 15
4. 9 + B = 15
B = 15 - 6
B = 15 - 9
B=9
B=6
Memorize
this type of
problem so
you can
solve using
the same
operation
5. 6 x R = 48
48  6 = R
8=R
6. W x 8 = 48
48  8 = w
6=w
Find the
example
that does not
use the inverse
Operation.
7. 72  S = 9 Memorize
this type of
72  9 = S
problem so
8=S
you can
solve using
the same
operation
8. M  9 = 8
8x9=M
72 = M
Equations are like a Balance scale. If weight is added or subtracted to
one side it will make the scale tip or not be balanced. In order to keep the
scale balanced you must add or subtract equal amounts to both sides.
This idea will help us solve equations with variables.
Property of Equality for Addition and Subtraction
F - 244 =120
F - 244 + 244 = 120 + 244
Solve
number to each side of an equation, then the 2 sides
remain the same.
F = 364
X + 3.6 = 12.4
Subtract 3.6 from both sides
X + 3.6 - 3.6 = 12.4 - 3.6
Solve
X = 8.8
Subtraction Property of Equality If you subtract the
same number from each side of an equation, then the 2
sides remain the same.
Property of Equality for Multiplication and Division
434 = 2s
434 = 2s
2
2
Remember a variable next
to a number indicates
multiplication
S = 217
Divide both sides by 2
Solve
Division Property of Equality If each side of an
equation is divided by the same nonzero number, then
the 2 sides remain equal.
n  8 = 96
Multiply both sides by 8
n  8 x 8 = 96 x 8
Solve
n = 768
Multiplication Property of Equality If each side of an
equation is multiplied by the same number, then the 2
sides remain equal.
An expression is one part of an equation
Examples
15 - 6 20 + a
Numerical expressions are often
written in sentence form.
7-x
42  d
Five more hits than the Yankees Ten fewer points than the Knicks
Yankees + 5
Knicks - 10
or
Y+5
K - 10
The key words often indicate what to do
fewer means subtract
Multiplication
Plus
Times
Sum
Product
Subtraction
multiplied
More than
Minus
Increased by
Difference
Division
Total
Less than
Divided
Subtract
quotient
Decreased by
3x = 18
3x + 9 = 18
How are these equations different?
When one side of an equation has 2 or more operations we need
more than one step to solve.
3x + 9 = 18
First subtract 9 from both sides.
3x + 9 - 9 = 18 - 9
Simplify
3x = 9
Then divide each side by 3
3x = 9
3
3
So x = 3
B - 0.8 = 1.3
9
First add 0.8 to both sides
B - 0.8 + 0.8 = 1.3 + 0.8
9
Simplify
B = 2.1
9
B x 9 = 2.1 x 9
9
Then multiply both sides by 9
So B = 18.9
An integer is a whole number that can be either greater than 0, called positive,
or less than 0, called negative. Zero is neither positive nor negative.
Two integers that are the same distance from zero in opposite directions are called
opposites.
-5 is the opposite of 5
Every integer on the number line has an absolute value,
which is its distance from zero. The brackets indicate the absolute value.
Using a Number Line to Add or Subtract Integers
Add a positive integer by moving to the right on the number line
Add a negative integer by moving to the left on the number line
2+6=?
8 + (- 3 ) = ?
2
6
8
-3
3-7=?
Subtract an integer by adding its opposite
-7
3
3 - (-7) = ?
7
3
Another way to remember the rule for
subtraction is Keep, change, change!
3- 7=?
Keep the first number
Change the sign
+
3
Change the
sign of the
second number
-7
4-9=?
Keep
the 4
4
+
(-9)
Change the
sign
Change the sign
of the 9
To multiply or divide signed integers, always multiply or divide the absolute values
and use these rules to determine the sign of the answer:
If the signs are the same the product or quotient is Positive
6 x 3 = 18
(-6 ) x (-3) = 18
If the signs are different the product or quotient is
Negative
(-6 ) x 3 = -18
6 x (-3) = -18
Using Counters to Solve Integer Problems
A positive and a negative counter equal 0
+
_
To add we put counters in the box
+
+
_
+
_
+
We put 3 negative counters in
_
The positive and negative pairs cancel each
other out to make 0
We are left with 1 positive counter
Subtracting Integers With Counters
To subtract we take counters out of the box
4 - (-5)
+
+
_
+
+
_
+
+
_
+
_
+
+
_
Since there are no negative counters we
to the box we do not change the value
If we take out the 5 negatives we
will be left with 9 positives
+
+
+
+
+
+
+
+
+
4 - (-5) = 9
+
_
Using Counters to Multiply Integers
In the sentences 3 x 2 we will place 3 groups of 2 in the box
+
+
+
+
+
+
+
+
+
+
+
+
Multiplying a Positive Integer by a Negative Integer
In the problem 3 x (-2) we are putting in 3 groups of 2 negatives
in the box
_
_
_
_
_
_
_
_
_
_
_
_
The box shows that 3 x (-2) = - 6
Multiplying With a Negative Integer
When multiplying by a negative integer we are taking out groups
In the problem (-2 ) x 3 we are
taking out 2 groups of 3
Since the box is empty we must
add 0 pairs until we have enough
to take out 2 groups of 3
+
_
+
_
+
_
+
_
+
_
+
_
+
_
+
_
+
_
+
_
+
_
+
_
We can now take out 2 groups of 3
(-2) x 3 = -6
_
_
_
_
_
_
We are left with
6 negatives
3x + 9 = 18
3x + 9 > 18 How are these number sentences different?
When an equation has > or < sin it is called an inequality
We solve inequalities in the same way as equations
3x + 9 > 18
First subtract 9 from both sides.
3x + 9 - 9 > 18 - 9
Simplify
3x > 9
Then divide each side by 3
3x > 9
3
3
So x > 3
We can graph the solution to an inequality on a number line.
2a - 5 > 9
First, solve as you would an equation
2a - 5 + 5 > 9 + 5
2a > 9
2
2
a > 2.5
Divide both sides by 2
Since a is greater than 2.5 it will include all
numbers greater than 2.5 but not 2.5
We can draw a circle at 2.5 to show this.
coordinate plane
The plane determined by a horizontal number line, called the x-axis,..
and a vertical
number line,
called the yaxis,
Each point in the
coordinate plane
can be specified
by an ordered
pair of numbers
(-3,1)
intersecting at a
point called the
origin
Here's one way geometry is
used in the real world. A team
of archaeologists is studying the
ruins of Lignite, a small mining
town from the 1800's. They plot
points on a coordinate plane to
show exactly where each
artifact is found.
(1,3)
Name each
Point.
(-5,2) (-2,4)
(3,-4) (5,-6)
(5,-4)
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